# Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $$F,G$$ to be $$dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$$. Then we proved that it satisfies the 7 Properties for intersection numbers (I am not sure if these are well known or not, but I mean the ones from William Fultons book about algebraic curves).
My question is regarding the intersection number of two projective curves $$F,G$$, which we defined by $$dim_k (\mathcal{O}_P (\mathbb{P}^2)/(F_*,G_*))$$, where $$F_*$$ and $$G_*$$ are the corresponding dehomogenized forms. Our professor only told us that this satisfies the 7 properties again (however you have to change 2 of them a little so that they make sense in the projective case). Now I wanted to know if you actually have to check every single one of the properties again, or if there is a short argument tracing back to the affine case.